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Process level moderate deviations for stabilizing functionals

Published online by Cambridge University Press:  07 October 2008

Peter Eichelsbacher  and
Tomasz Schreiber
Peter Eichelsbacher
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, NA 3/68, 44780 Bochum, Germany; [email protected]
Tomasz Schreiber
Affiliation:
Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Toruń, Poland; [email protected]
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Abstract


Functionals of spatial point process often satisfy a weak spatial dependencecondition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which isa level-3 result for empirical point fields as well as a level-2 resultfor empirical point measures. The level-3 rate function coincides withthe so-called specific information. We show that the general resultcan be applied to prove MDPs for various particular functionals,including random sequential packing, birth-growth models, germ-grainmodels and nearest neighbor graphs. 


Keywords

Moderate deviationsrandom Euclidean graphsrandom sequential packing.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 1 - 15
Copyright
© EDP Sciences, SMAI, 2010

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References

Baryshnikov, Y. and Yukich, J.E., Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213–253. CrossRef
Baryshnikov, Y., Eichelsbacher, P., Schreiber, T. and Yukich, J.E., Moderate Deviations for some Point Measures in Geometric Probability. Ann. Inst. H. Poincaré 44 (2008) 422–446; electronically available on the arXiv, math.PR/0603022. CrossRef
Comets, F., Grandes déviations pour des champs de Gibbs sur ${\Bbb Z}^d$ (French) [ Large deviation results for Gibbs random fields on ${\Bbb Z}^d$ ] . C. R. Acad. Sci. Paris Sér. I Math. 303 (1986) 511–513.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Second edition. Springer (1998).
Föllmer, H. and Orey, S., Large Deviations for the Empirical Field of a Gibbs Measure. Ann. Probab. 16 (1988) 961–977. CrossRef
H.-O. Georgii, Large Deviations and Maximum Entropy Principle for Interacting Random Fields on ${\Bbb Z}^d.$ Ann. Probab. 21 (1993) 1845–1875.
Georgii, H.-O., Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields 99 (1994) 171–195. CrossRef
Georgii, H.-O. and Zessin, H., Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96 (1993) 177–204. CrossRef
P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988).
I.S. Molchanov, Limit Theorems for Unions of Random Closed Sets. Lect. Notes Math. 1561. Springer (1993)
Large De, S. Ollaviations for Gibbs Random Fields. Probab. Theor. Rel. Fields 77 (1988) 343–357.
Penrose, M.D., Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005) 1945–1991. CrossRef
M.D. Penrose, Gaussian Limits for Random Geometric Measures, Electron. J. Probab. 12 (2007) 989–1035. CrossRef
Penrose, M.D. and Yukich, J.E., Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005–1041.
Penrose, M.D. and Yukich, J.E., Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272–301.
Penrose, M.D. and Yukich, J.E., Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277–303.
A. Rényi, Théorie des éléments saillants d'une suite d'observations, in Colloquium on Combinatorial Methods in Probability Theory. Mathematical Institut, Aarhus Universitet, Denmark (1962), pp. 104–115.
D. Stoyan, W. Kendall and J. Mecke, Stochastic Geometry and Its Applications. Second edition. John Wiley and Sons (1995).